SIMPLE ZEROS OF THE RIEMANN ZETA-FUNCTION

J. B. Conrey, A. Ghosh, S. M. Gonek

1. Introduction and notation.

Let ζ(s) denote the Riemann zeta-function and ρ = β + iγ a zero of ζ(s). In the process of evaluating N(T), the number of zeros of ζ with 0 < γ ≤ T, each zero is counted with multiplicity, denoted by m(ρ), to give the von Mangoldt formulae

N(T) = \frac{T}{2π} log \frac{T}{2πe} + \frac{7}{8} + S(T) + O (\frac{1}{T} ) ,

S(T) = \frac{1}{π} arg ζ(\frac{1}{2} + iT) ≪ log T.

It is not known if m(ρ) = 1 for all ρ but it is so conjectured. Indeed, all the zeros located so far, more than 1.5 × 10^{9} of them, are simple. Support for this conjecture is further strengthened by the fact that Montgomery’s pair correlation conjecture (see [M2]) implies that almost all the zeros are simple.

It is known that at least two-fifths of the zeros are simple. (See Conrey [C2],

[C3] for a discussion of results in this direction). Assuming the Riemann-Hypothesis (RH), Montgomery [M2] has shown that at least two-thirds of the zeros are simple, and later, with Taylor [M3], the constant two-thirds was improved to

\frac{3}{2} − \frac{\sqrt{2}}{2} cot \frac{\sqrt{2}}{2} = 0.6725 ….

Cheer and Goldston [ChG] have recently improved upon this result slightly by showing that RH implies that at least 0.672753 of the zeros are simple.

A problem related to the simplicity of the zeros of ζ(s) is the enumeration of distinct zeros (that is to count each zero precisely once without regard to its multiplicity).

Let us define

N_{d}(T) = |{ρ = β + iγ : 0 < γ ≤ T, ζ(ρ) = 0}|;
N_{s}(T) = |{ρ = β + iγ : 0 < γ ≤ T, ζ(ρ) = 0, m(ρ) = 1}|.

We are interested in getting bounds for the constants C_{d} and C_{s} defined by

C_{d} = lim inf_{T →∞} N_{d}(T)/N(T)

Typeset by AMS-TEX

Supported in part by an NSF grant

C_{s} = lim inf_{T →∞} N_{s}(T)/N(T).

Of course, it is clear that C_{d} = 1 if and only if C_{s} = 1. To get a bound for C_{s}, Montgomery proceeded by observing that

N_{s}(T) ≥ ∑_{γ≤T}(2 − m(γ)) = 2N(T) − ∑_{γ≤T} m(γ) ,

(where all sums over zeros are repeated according to multiplicity and we use m(γ) in place of m(ρ) on assuming RH), and he showed in [M2] that

∑_{γ≤T} m(γ) ≤ (C_{0} + o(1))N(T) (C_{0} = \frac{4}{3}, T → ∞).

One gets a bound for C_{d} by observing that

2N_{s}(T) ≤ ∑_{γ≤T} \frac{(m(γ) − 2)(m(γ) − 3)}{m(γ)} = ∑_{γ≤T} m(γ) − 5N(T) + 6N_{d}(T),

so that

C_{d} ≥ \frac{5 + 2C_{s} − C_{0}}{6} .  (1.1)

Due to the relationship between C_{0} and C_{s} in Montgomery’s method, we get

C_{d} ≥ \frac{9 − 3C_{0}}{6} ≥ \frac{5}{6}  if C_{0} = \frac{4}{3}.  (1.2)

(The constants are improved very slightly by using the better value for C_{0} in [M3] or in [ChG]). If one could devise a method for evaluating C_{s} independently of C_{0}, then (1.1) would be essentially better than (1.2).

In this paper we develop a new method to obtain lower bounds for C_{s} and C_{d}. Our method requires the assumption of the Riemann Hypothesis as well as an assumption of an upper bound for averages of sixth moments of Dirichlet Lfunctions L(s, χ). This sixth moment hypothesis is implied by the Generalized Lindelöf Hypothesis, which is the assumption that for any ε > 0,

L(s, χ) ≪_{ε} (q(1 + |t|))^{ε}

for σ ≥ \frac{1}{2}, where χ is a character mod q. The Generalized Lindelöf Hypothesis is weaker than the Generalized Riemann Hypothesis which conjectures that L(s, χ) ≠ 0 if σ > \frac{1}{2}. For convenience, we state our main result as

Theorem 1. Assuming the Riemann Hypothesis (RH) and the Generalized Lindelof Hypothesis (GLH),

C_{s} ≥ \frac{19}{27}, C_{d} ≥ \frac{5}{6} + \frac{1}{81}.

Remark. In an earlier version of this paper, we had omitted the assumption of GLH. We thank the referee for pointing out this mistake. Also our paper [CGG], which refers to the methods here, requires this additional assumption of GLH.

1.1 Notation.

We shall use ∗ to denote Dirichlet convolution of arithmetic functions. Thus if α and β are arithmetic functions, then

(α ∗ β)(n) = ∑_{d|n} α(d)β (\frac{n}{d}) .

Further, we define operators T_{s} and L^{k} by

(T_{s}α)(n) = n^{s}α(n);
(L^{k}β)(n) = (log n)^{k}β(n).

Note that

T_{s}(α ∗ β) = (T_{s}α) ∗ (T_{s}β)

and

L(α ∗ β) = (Lα) ∗ β + α ∗ (Lβ).

Also, observe that if k is squarefree and α is multiplicative (i.e., α(mn) = α(m)α(n) wherever (m, n) = 1), then

α(β ∗ γ)(k) = (αβ ∗ αγ)(k).

We use τ_{r} to denote the r-fold divisor function (often denoted d_{r}( )). Often this has an unspecified but bounded positive integer subscript r, not necessarily the same at each occurrence. We make frequent use of the inequalities τ_{r}(n) τ_{s}(n) ≤ τ_{rs}(n) and τ_{r}(mn) ≤ τ_{r}(m)τ_{r}(n) for positive integers r, s, m and n. We use I for the arithmetic function which is the identity for Dirichlet convolution, i.e., I(1) = 1 and I(n) = 0 for n > 1. Also, we use 1 for the arithmetic function for which 1(n) = 1 for all n.

Throughout L shall denote log \frac{T}{2π}, and e(x) denotes e^{2πix}.

2. Sketch of the proof for C_{s}

We note that ρ is a simple zero of ζ(s) if and only if ζ′(ρ) ≠ 0. By Cauchy’s inequality, it follows that

|∑_{γ≤T} Bζ′(ρ)|^{2} ≤ N_{s}(T) ∑_{γ≤T} |Bζ′(ρ)|^{2},  (2.1)

where B(s) is any regular function. We shall take

B(s) = ∑_{k≤y} b(k)k^{−s},  (2.2)

where

y = T^{θ}  (2.3)

and

b(k) = μ(k)P (\frac{log y/k}{log y} )  (2.4)

where P (x) is a polynomial with real coefficients which satisfies

P (0) = 0, P (1) = 1.

Then, if θ < \frac{1}{2}, we shall show, Theorem 2. Assuming GLH, we have,

S_{1} := ∑_{γ≤T} Bζ′(ρ) ∼ (\frac{1}{2} + θ ∫_{0}^{1} P (x) dx) \frac{T L^{2}}{2π}  (2.5)

and

S_{2} := ∑_{γ≤T} Bζ′(ρ)Bζ′(1 − ρ) ∼ λ\frac{T L^{3}}{2π} ,  (2.6)

where

λ = \frac{1}{3} + (θ ∫_{0}^{1} P (x) dx)^{2} + θ ∫_{0}^{1} P (x) dx + \frac{1}{12θ} ∫_{0}^{1} P′(x)^{2} dx.  (2.7)

Assuming the Riemann Hypothesis, S_{2} = ∑_{γ≤T} |Bζ′(ρ)|^{2}. (This is the only place

we need RH.) Our estimate for C_{s} is then derived by choosing the polynomial P (x). By the calculus of variations, the optimal choice is

P (x) = −θx^{2} + (1 + θ)x.

Then

∫_{0}^{1} P (x) dx = \frac{1}{2} + \frac{θ}{6}, ∫_{0}^{1} P′(x)^{2} dx = \frac{θ^{2}}{3} + 1

so that with θ = \frac{1}{2} − ε (ε → 0^{+}) we obtain

S_{1} ∼ \frac{19}{24}N(T)L, S_{2} ∼ \frac{57}{64}N(T)L^{2}

and Theorem 1 follows.

To obtain the formulae (2.5) and (2.6), we use Cauchy’s residue theorem to write

S_{1} = \frac{1}{2πi} ∫_{C} \frac{ζ′}{ζ} (s)ζ′(s)B(s) ds  (2.8)

and

S_{2} = \frac{1}{2πi} ∫_{C} \frac{ζ′}{ζ} (s)ζ′(s)ζ′(1 − s)B(s)B(1 − s) ds,  (2.9)

where C is the positively oriented rectangle with vertices at 1 − c + i, c + i, c + iT, and 1 − c + iT, and where

c = 1 + L^{−1}  (2.10)

and T is chosen so that the distance from T to the nearest γ is ≫ L^{−1}. (It is easy to see that this assumption on T involves no loss of generality.)

Remark. There is a fair amount of work involved in evaluating S_{1} and S_{2}, even if one considers only main terms without regard to error terms. Therefore, let us motivate this work by a suggestion as to what we might expect, by considering averages over all t instead of just ordinates of zeros of ζ(s). In this case we would start with the function G(s)B(s) instead of ζ′(s)B(s) where (see (3.2) for the definition of χ(s))

G(s) = ζ′(s) − \frac{χ′}{χ} (s)ζ(s)

and for P we take P (x) = x. Note that G and ζ′ agree at s = ρ. Then it is trivial to see that

∫_{1}^{T} GB(\frac{1}{2} + it) dt ∼ T L

by writing the integral as a complex integral and moving the line of integration to σ = c > 1. Also, we know that

∫_{1}^{T} |GB(\frac{1}{2} + it)|^{2} dt ∼ \frac{4}{3}T L^{2}

since Levinson evaluates this integral in his work [L]. (He considers the integral more generally on the a-line where \frac{1}{2} − a ≪ (log T)^{−1}.) Thus, the consideration of integral means leads one to suspect that our method might lead to a proof that \frac{3}{4} of the zeros are simple. However, one can see from the formulae (2.5) and (2.6) that the average at zeros is slightly different from the average over all t.

3. Preliminary Transformations.

The ensuing analysis is much simpler if, instead of (2.8) and (2.9), we use slightly different expressions.

The functional equation for ζ(s) is given by

ζ(s) = χ(s)ζ(1 − s)  (3.1)

where

χ(1 − s) = χ(s)^{−1} = 2(2π)^{−s}Γ(s) cos \frac{πs}{2}.  (3.2)

We differentiate to obtain

ζ′(s) = −χ(s) (ζ′(1 − s) − \frac{χ′}{χ} (s)ζ(1 − s)) .  (3.3)

We first consider S_{1}. It follows from (2.5) and (3.3) that

S_{1} = − ∑_{γ≤T} χ(ρ)ζ′(1 − ρ)B(ρ)

= \frac{1}{2πi} ∫_{C} \frac{ζ′}{ζ} (1 − s)χ(s)ζ′(1 − s)B(s) ds.  (3.4)

Let s = σ + it be on C. We shall use the following well-known estimates:

χ(s) ≪ |t|^{\frac{1}{2} −σ},  (3.5)
B(s) ≪ y^{1−σ}L,  (3.6)

ζ′(1 − s) ≪ |t|^{σ/2}L^{2},  (3.7)

and

\frac{ζ′}{ζ} (1 − s) ≪ L^{2}  (3.8)

for s on C; the last estimate holds if the distance from t to the nearest γ is ≫ L^{−1} which we have assumed. Hence, the contribution to S_{1} from the integral in (3.4) taken along the horizontal sides of C is

≪ T^{c/2}y^{c}L^{5} ≪ T^{\frac{1}{2}} yL^{5}.  (3.9)

To estimate the contribution to S_{1} from the right side of the contour C, we use (3.3) to replace χ(s)ζ′(1 − s) in (3.4). Also, by (3.1) and (3.2),

\frac{ζ′}{ζ} (s) + \frac{ζ′}{ζ} (1 − s) = \frac{χ′}{χ} (s) = − log \frac{|t|}{2π} + O (\frac{1}{1 + |t|}) .  (3.10)

Thus, the integral along the right side of C is

= \frac{1}{2πi} ∫_{c+i}^{c+iT} (\frac{χ′}{χ} (s)^{2}ζ(s) − 2\frac{χ′}{χ} (s)ζ′(s) + \frac{ζ′}{ζ} (s)ζ′(s)) B(s) ds

= \frac{T}{2π} L^{2} + O(T L),  (3.11)

since only the term n = 1 from the Dirichlet series for ζ(s)B(s) contributes anything to the main term.

To evaluate the contribution to S_{1} from the left side of the contour C, we make the change of variable s → 1 − s in (3.4). Then, the integral along the left side is equal to −\overline{I}_{1} where

I_{1} = \frac{1}{2πi} ∫_{c+i}^{c+iT} χ(1 − s)\frac{ζ′}{ζ} (s)ζ′(s)B(1 − s) ds.  (3.12)

So far, we have shown by (3.4), (3.9), (3.11), and (3.12) that

S_{1} = \frac{T}{2π} L^{2} − \overline{I}_{1} + O(T L) + O(T^{\frac{1}{2}} yL^{5}).  (3.13)

We now consider S_{2}. By (2.6), (3.3), and Cauchy’s theorem,

S_{2} = − ∑_{γ≤T} χ(1 − ρ)ζ′(ρ)^{2}B(1 − ρ)B(ρ)

= −\frac{1}{2πi} ∫_{C} χ(1 − s)\frac{ζ′}{ζ} (s)ζ′(s)^{2}B(s)B(1 − s) ds.  (3.14)

By the estimates (3.5) – (3.8), the contribution to S_{2} of the integral over the horizontal sides of the contour is

≪ T^{\frac{1}{2}} yL^{8}.  (3.15)

The integral along the left side of C is, by (3.3) and (3.10),

= \frac{1}{2πi} ∫_{1−c+i}^{1−c+iT} (\frac{χ′}{χ} (s) − \frac{ζ′}{ζ} (1 − s)) χ(s)

× (\frac{χ′}{χ} (s)ζ(1 − s) − ζ′(1 − s))^{2} B(s)B(1 − s) ds

which, by the change of variable s → 1 − s and another use of (3.3), is

= \frac{−1}{2πi} ∫_{c−1}^{c−iT} χ(1 − s) (\frac{χ′}{χ} (s)^{3}ζ(s)^{2} − 3\frac{χ′}{χ} (s)ζ′(1 − s)χ(s)

−\frac{ζ′}{ζ} (s)ζ′(s)^{2}) B(s)B(1 − s) ds.  (3.16)

We multiply this out and rewrite it as a sum of three integrals. For the first two of these we move the line of integration to the \frac{1}{2}-line with the same error term as in (3.15). Thus the above is

= M_{1} + M_{2} − \overline{I}_{2} + O(T^{\frac{1}{2}} yL^{8})  (3.17)

where

M_{1} = \frac{1}{2π} ∫_{1}^{T} \frac{χ′}{χ} (\frac{1}{2} − it)^{3}|Bζ(\frac{1}{2} + it)|^{2}dt,  (3.18)

M_{2} = −\frac{3}{2π} ∫_{1}^{T} \frac{χ′}{χ} (\frac{1}{2} − it)|Bζ′(\frac{1}{2} + it)|^{2} dt,  (3.19)

and

I_{2} = \frac{1}{2πi} ∫_{c+i}^{c+iT} χ(1 − s)\frac{ζ′}{ζ} (s)ζ′(s)^{2}B(s)B(1 − s) ds.  (3.20)

The integral in (3.14) taken along the right side of C is equal to −I_{2}. Thus, by (3.15) and (3.17) , we see that

S_{2} = M_{1} + M_{2} − 2ℜI_{2} + O(T^{\frac{1}{2}} yL^{8}).  (3.21)

The integrals in M_{1} and M_{2} have been evaluated (implicitly) in Conrey [C1]. Let

F (s) = ∑_{n=0}^{N}(−1)^{n}a_{n}ζ^{(n)}(s)L^{−n} = ∑_{m=1}^{∞} Q (\frac{log m}{L}) m^{−s}  (3.22)

where

Q(x) = ∑_{n=0}^{N} a_{n}x^{n}

is a polynomial with real coefficients. Let

M = ∫_{T}^{T +U} |BF (\frac{1}{2} + it)|^{2} dt  (3.23)

where

U = T L^{−10}

and B(s) is as in (2.2). Then

M = U (Q(0)^{2} + θ ∫_{0}^{1} ∫_{0}^{1} (P (u)Q′(v) + \frac{1}{θ}P′(u)Q(v))^{2} du dv)

+O (UL^{−1}(log L)^{5}) ,  (3.24)

provided θ < \frac{1}{2}. Moreover, for T ≤ t ≤ T + U, we have

\frac{χ′}{χ} (\frac{1}{2} − it) = −L + O(L^{−10}).  (3.25)

Thus, we can replace χ′/χ by −L and sum integrals of length U to obtain estimates for M_{1} and M_{2}. For M_{1} we take Q_{1}(x) = 1, and for M_{2} we take Q_{2}(x) = −x. Thus, we get

M_{1} = \frac{−T L^{3}}{2π} (1 + \frac{1}{θ} ∫_{0}^{1} P′(u)^{2} du) + O (T L^{2}(log L)^{5})  (3.26)

and

M_{2} = \frac{3T L^{3}}{2π} (\frac{1}{2} + θ ∫_{0}^{1} P (u)^{2}du + \frac{1}{3θ} ∫_{0}^{1} P′(u)^{2} du)

+O(T L^{2}(log L)^{5}).  (3.27)

We have shown that (3.26) and (3.27) are valid for θ < \frac{1}{2} as T → ∞. It is worth remarking that (3.26) and (3.27) are valid for θ < \frac{4}{7}, by the work in [C3].

It remains to estimate I_{1} and I_{2} of (3.12) and (3.20). Indeed, this forms the main area of difficulty and requires a bit of preparation, in the form of lemmas, which we state in the next section.

4. Preliminary Lemmas.

Lemma 1. Let r > 0. Then, for any c_{0} > 0,

\frac{1}{2πi} ∫_{c+i}^{c+iT} χ(1 − s)r^{−s} ds = \{ \begin{matrix}e^{−2πir} + E(r, c)r^{−c} & if r ≤ \frac{T}{2π} \\ E(r, c)r^{−c} & otherwise, \end{matrix}  (4.1)

uniformly for c_{0} ≤ c ≤ 2 where

E(r, c) ≪ T^{c− \frac{1}{2}} + \frac{T^{c+ \frac{1}{2}}}{|T − 2πr| + T^{\frac{1}{2}}} .  (4.2)

This is provided implicitly in Gonek ([Go]; Lemma 2).

SIMPLE ZEROS OF THE RIEMANN ZETA-FUNCTION 9

Lemma 2. Suppose that A(s) = ∑_{n=1}^{∞} a(n)n^{−s} for σ > 1 where

a(n) ≪ τ_{k_{1}}(n)(log n)^{ℓ_{1}}

for some non-negative integers k_{1} and ℓ_{1}. Let B(s) = ∑_{n≤y} b(n)n^{−s} where

b(n) ≪ τ_{k_{2}}(n)(log n)^{ℓ_{2}}

for non-negative integers k_{2} and ℓ_{2} and where

T^{ϵ} ≪ y ≪ T

for some ϵ > 0. Let

I = \frac{1}{2πi} ∫_{c+i}^{c+iT} χ(1 − s)B(1 − s)A(s) ds

where c = 1 + 1/ log T. Then

I = ∑_{n≤y} \frac{b(n)}{n} ∑_{m≤ \frac{nT}{2π}} a(m)e(−m/n) + O (T^{1/2}y(log T)^{A})

for some A.

Remark. An admissible value for A is k_{1} + k_{2} + ℓ_{1} + ℓ_{2}. Proof. By Lemma 1 it suffices to show that

∑_{n≤y} |b(n)| ∑_{m} \frac{|a(m)|}{m^{c}} (T^{1/2} + f_{m,n}(T)) ≪ yT^{1/2}(log T)^{A}

where

f_{m,n}(T) = \frac{T^{3/2}}{|T − 2πm/n| + T^{1/2}} .

Now by a Theorem of Shiu [S], for any ϵ > 0,

∑_{X−Y <n≤X} τ_{k}(n)(log n)^{ℓ} ≪_{ϵ} Y (log X)^{k+ℓ−1}

provided that Y ≫ X^{ϵ}. It follows easily that

∑_{n≤y} |b(n)| ∑_{m} \frac{|a(m)|}{m^{c}} ≪ y(log T)^{A}

with A = k_{1} +k_{2} +ℓ_{1} +ℓ_{2} −1. Thus, the term with the T^{1/2} is acceptable. Next, we break up the sum involvingf_{m,n} into three parts. The terms with |T −2πm/n| > T/2 have f_{m,n}(T) ≪ T^{1/2} so this case is like the one we have just discussed. Now consider the terms for which

T^{1/2} ≤ |T − 2πm/n| ≤ T/2.

Assume first of all that

T^{1/2} ≤ 2πm/n − T ≤ T/2,

the other inequality leading to a similar argument. We further refine the sum into ≪ log T sums of the shape

T + P < \frac{2πm}{n} < T + 2P

where T^{1/2} ≪ P ≪ T. For m and n satisfying this condition, f_{m,n}(T) ≪ P^{−1}. Also, m ≈ nT and m ranges over an interval of length ≈ nP so that the contribution from one of these sums is

≪ T^{3/2} ∑_{n≤y} |b(n)| ∑_{m} \frac{|a(m)|}{nT}P^{−1}

which, by Shiu’s theorem, is

≪ yT^{1/2}(log T)^{A}

for some A. Summing this over the ≪ log T values of P contributes an additional factor of log T.

Finally, we consider the contribution of the terms for which

|T − 2πm/n| ≤ T^{1/2}.

For such values of m and n we have f_{m,n}(T) ≪ T^{−1/2}. For a given n the admissible values of m are ≈ nT in size and range over an interval of length ≈ nT^{1/2}. Thus, Shiu’s theorem is applicable and we find that the total contribution from these terms is

≪ T^{3/2} ∑_{n≤y} |b(n)| ∑_{m} \frac{|a(m)|}{nT}T^{−1/2}

≪ yT^{1/2}(log T)^{A}

for some A. This concludes the proof.

Lemma 3. Suppose that

A_{j}(s) = ∑_{m=1}^{∞} a_{j}(m)m^{−s}

and that

A(s) = ∑_{m=1}^{∞} a(m)m^{−s} = ∏_{j=1}^{J} A_{j}(s),

for some natural number J ≥ 1. Then, for any integer d > 0, and any completely multiplicative function f,

∑_{m=1}^{∞} f(m)a(md)m^{−s} = ∑_{d=d_{1}···d_{J}}  ∏_{j=1}^{J} ∑_{(m,d_{1}...d_{j−1})=1}^{∞} f(m)a_{j}(md_{j})m^{−s}.

Proof. The assertion for J > 2 follows from J − 1 applications of the case J = 2, which we now prove. This case follows immediately on observing that for any two arithmetical functions α and β,

(α ∗ β)(md) = ∑_{\begin{matrix}g|m \\ (h,m/g)=1 \end{matrix}} ∑_{h|d} α(gh)β (\frac{md}{gh});

for the right-hand side is equal to

∑_{ℓ|md} α(ℓ)β (\frac{md}{ℓ}) ∑_{\begin{matrix}gh=ℓ \\ (h,m/g)=1 \end{matrix}} 1 = ∑_{ℓ|md} α(ℓ)β (\frac{md}{ℓ}) ∑_{\begin{matrix}g|ℓ \\ (ℓ,m)=g \end{matrix}} 1,

which is the same as the left hand side. □

Lemma 4. (Perron’s formula). Suppose that A(s) = a(n)n^{−s} for σ > 1 where

|a(n)| ≤ Kτ_{k}(n)(log n)^{ℓ}.

Then, for any ε > 0,

R = ∑_{n≤x} a(n) − \frac{1}{2πi} ∫_{c−iU}^{c+iU} A(s)\frac{x^{s}}{s} ds <<_{ε} Kx^{ε}(1 + xU^{−1})

where c = 1 + (log x)^{−1} and the implied constant is independent of K. If U ≪ x^{1−ε} for some ε > 0, then

R ≪_{ε} KxU^{−1}(log x)^{k+ℓ}.

Proof. By the lemma of Section 17 of Davenport [D],

R ≪ ∑_{\begin{matrix}n=1 \\ n≠x \end{matrix}} (\frac{x}{n})^{c} |a(n)|min(1, U^{−1}| log x/n|^{−1}) + |a(x)|,

the last term occurring only if x is an integer. The lemma follows easily from this, (4.4), and the fact that a(n) ≪_{ε} n^{ε} for any ε > 0. □

12 J. B. CONREY, A. GHOSH, S. M. GONEK

Lemma 5. Suppose that y, d ≪ T with log y ≫ log T. Let

Σ(d, M) := ∑_{k≤y/d} \frac{b(kd)}{k}(log k)^{M}.

Then for any ε > 0,

Σ(d, M) = \{ \begin{matrix}(−1)^{M} \frac{μ(d)d}{ϕ(d)}P^{(1−M)} (\frac{log y/d}{log y}) (log y)^{M−1} + O_{ε}(E_{M}) & if 0 ≤ M ≤ 1 \\ O_{ε}(E_{M}) & if M ≥ 2 \end{matrix}

where with

b = (C log L)^{−1}

for some constant C > 0 and

F_{1}(d) = ∏_{p|d}(1 + p^{−3/4}).

In all cases,

Σ(d, M) ≪_{ε} L^{M−1+ε}F_{1}(d).

This follows from Lemma 10 of Conrey [C1]. Lemma 6. If Q is a polynomial, then

∑_{d≤y} \frac{μ^{2}(d)}{ϕ(d)}Q (\frac{log y/d}{log y}) = (∫_{0}^{1} Q(u) du) log y + O(1).

This is well-known in the case Q(x) ≡ 1 and the general case may be deduced from this particular one. Lemma 7. Let F_{1}(d) = ∏_{p|d}(1 + p^{−3/4}). If r ≤ 4, then

∑_{d≤y} \frac{F_{1}(d)^{r}}{d} ≪ log y

and

∑_{d≤y} \frac{F_{1}(d)^{r}}{d^{θ}} ≪ \frac{y^{1−θ}}{1 − θ}

uniformly for \frac{3}{4} < θ < 1. Proof. We have

F_{1}(d)^{r} = ∏_{p|d}(1 + p^{−3/4})^{r} ≪ ∏_{p|d}(1 + 4p^{−3/4})

≪ ∑_{n|d} \frac{τ(n)^{2}}{n^{3/4}} ≪ ∑_{n|d} n^{−1/2},

whence

∑_{d≤y} \frac{F_{1}(d)^{r}}{d^{θ}} ≪ ∑_{m≤y} m^{−θ} ∑_{n≤y/m} \frac{1}{n\frac{1}{2} + θ} ≪ ∑_{m≤y} m^{−θ}.

The result follows. □

Lemma 8. Let A be an integer, 1 ≤ A ≤ 4, and let P be a polynomial with P (0) = 0. For x ≤ y define

S = S(x, A, d) = ∑_{\begin{matrix}g≤x \\ (g,d)=1 \end{matrix}} \frac{Λ(g)(log g)^{A}}{ϕ(g)} P (\frac{log x/g}{log y} ) .

Then

S(x, A, d) = A!P^{(−A−1)} (\frac{log x}{log y}) (log y)^{A+1} + O(F_{1}(d))

where P^{(0)}(t) = P (t) and P^{(−r)}(t) = ∫_{0}^{t} P^{(−r+1)}(u) du and where F_{1} is as in Lemma 7.

Proof. We have

S = ∑_{p≤x} \frac{(log p)^{A+1}}{p} P (\frac{log x/p}{log y}) + O (1 + ∑_{p|d} p^{−1}(log p)^{A+1}) .

The error term here is ≪ 1 + ∑_{p|d} p^{−3/4} ≪ F_{1}(d). By Stieltjes’ integration, the sum

here is

= ∫_{1}^{x} u^{−1}(log u)^{A}P (\frac{log x/u}{log y}) dθ(u)

where, by the Prime Number Theorem, for some C > 0,

θ(u) = ∑_{p≤u} log p = u + O(u exp(−C(log u)^{1/2})).

Thus, the integral is

= ∫_{1}^{x} u^{−1}(log u)^{A}P (\frac{log x/u}{log y}) du + O(1)

= ∫_{0}^{1}(1 − α)^{A}P (α \frac{log x}{log y}) dα(log x)^{A+1} + O(1).

The lemma now follows by integration by parts A times. □

Lemma 9. Suppose that b = (C log L)^{−1} and x ≪ T. For integer β ≥ 0 and x ≥ 2,

∑_{g≤x} Λ_{β}(g)g^{−1} ≪ (log x)^{β}

and

∑_{g≤x} Λ_{β}(g)g^{b−1} ≪ (log x)^{β}x^{b}

where Λ_{β} = μ ∗ L^{β}1 (see (7.1.7)).

Proof. The assertion is trivial for β = 0 since Λ_{0}(1) = 1 and Λ_{0}(1) = 0 if n > 1. Hence we assume β ≥ 1. Then the first assertion follows by induction from the easily proven formulae

Λ_{β} = Λ ∗ Λ_{β−1} + LΛ_{β−1},

∑_{g≤x} \frac{Λ_{1}(g)}{g} = ∑_{g≤x} \frac{Λ(g)}{g} ≪ log x,

and the second assertion is an immediate consequence of the first. □

Lemma 10. Suppose that Q is a squarefree positive integer. Then for bounded r and fixed σ, 0 < σ < 1, we have

∑_{d|Q} τ_{r}(d)d^{−σ} ≪ exp (C(log q)^{1−σ}(log log Q)^{−1})

and

∑_{d|Q} τ_{r}(d)d^{−1} ≪ (log log Q)^{C}

for some positive constant C which is independent of Q.

Proof. Since the summand is multiplicative, we have

log ∑_{d|Q} τ_{r}(d)d^{−σ} = ∑_{p|Q} log(1 + τ_{r}(p)p^{−σ})

≪ ∑_{p|Q} p^{−σ}.

If the primes p|Q are denoted p_{1}, ⋯, p_{n} and the first n primes are q_{1}, ⋯, q_{n}, then q_{n} ≤ 2 log Q if Q is large enough. Moreover, p_{i}^{−σ} ≤ q_{i}^{−σ} and by the Prime Number Theorem, the above sum is

≪ ∑_{q≤2 log Q} q^{−σ} ≪ ∫_{2^{−}}^{2 log Q} u^{−σ} dπ(u)

≪ \{ \begin{matrix}(log Q)^{1−σ}(log log Q)^{−1} & if 0 < σ < 1. \\ log log log Q & if σ = 1 \end{matrix}

The result now follows. □

5. Estimation of I_{1} and I_{2}.

By (3.12) and (3.20), we have

I_{ν} = \frac{1}{2πi} ∫_{c+i}^{c+iT} χ(1 − s)A_{ν}(s)B(1 − s) ds,  (5.1)

where ν = 1 and 2, with

A_{1}(s) = \frac{ζ′}{ζ} (s)ζ′(s) = ∑_{1}^{∞} a_{1}(m)m^{−s}  (5.2)

and

A_{2}(s) = \frac{ζ′}{ζ} (s)ζ′(s)^{2}B(s) = ∑_{1}^{∞} a_{2}(m)m^{−s}.  (5.3)

Thus, by Lemma 2 with k = 4 and ℓ = 3, we get I_{ν} = M_{ν} + Eν where

M_{ν} = ∑_{k≤y} ∑_{m≤ \frac{kT}{2π}} \frac{a_{ν}(m)b(k)}{k}e (−\frac{m}{k}) ,  (5.4)

and

E_{ν} ≪ T^{\frac{1}{2}} yL^{7}.  (5.5)

To evaluate M_{ν} we apply Perron’s formula to the sum over m and the main term arises from the residue of the pole at s = 1 of the generating function. If we assume the Generalized Riemann Hypothesis, then the error terms are easily handled. For our less conditional treatment, however, we must use Perron’s formula at a later stage after some preliminary rearrangements of the sum.

We first express the additive character e( ) in terms of multiplicative characters. Thus, if m/k = M/K where (M, K) = 1, then

e (−\frac{m}{k}) = e (−\frac{M}{K} ) = \frac{1}{ϕ(K)} ∑_{χ mod K} τ(\overline{χ})χ(−M)  (5.6)

where, as usual for a character χ mod K,

τ(χ) = ∑_{a=1}^{K} χ(a)e(a/K).  (5.7)

We wish to reduce the sum to a sum over primitive characters. If ψ mod q induces χ mod K, then (see [D], page 67)

τ(χ) = μ (\frac{K}{q} ) ψ (\frac{K}{q} ) τ(ψ).  (5.8)

Thus, by (5.7),

e (−\frac{m}{k}) = \frac{1}{ϕ(K)} ∑_{q|K} ∑_{ψ}^{∗}μ (\frac{K}{q} ) \overline{ψ} (\frac{K}{q} ) τ(\overline{ψ})ψ(−M)  (5.9)

since (M, K) = 1 implies that (M, q) = 1 whence χ(−M) = ψ(−M); here and elsewhere the ^{∗} indicates that the sum is over all primitive characters mod q. (The character mod 1 which induces all other principal characters will be included as a primitive character.) Finally, we wish to eliminate the dependence in our formula on g = g.c.d. (m, k). By the Möbius inversion formula, for any f,

f(M, K) = f (\frac{m}{g}, \frac{k}{g} ) = ∑_{d|g} ∑_{e|d} μ (\frac{d}{e}) f (\frac{m}{e}, \frac{k}{e} ) .  (5.10)

The condition d|g is equivalent to d|m, d|k. Thus,

\begin{aligned}e (−\frac{m}{k}) & = ∑_{\begin{matrix}d|m \\ d|k \end{matrix}} ∑_{e|d} \frac{μ(d/e)}{ϕ(k/e)} ∑_{q| \frac{k}{e}} ∑_{ψ}^{∗}μ (\frac{k}{eq} ) \overline{ψ} (\frac{k}{eq} ) τ(\overline{ψ})ψ (−\frac{m}{e}) \\ & = ∑_{q|k} ∑_{ψ}^{∗}τ(\overline{ψ}) ∑_{\begin{matrix}d|m \\ d|k \end{matrix}} ∑_{\begin{matrix}e|d \\ e| \frac{k}{q} \end{matrix}} \frac{μ(d/e)}{ϕ(k/e)}\overline{ψ} (\frac{−k}{eq} ) ψ (\frac{m}{e}) μ (\frac{k}{eq} ) \end{aligned}

= ∑_{q|k} ∑_{ψ}^{∗}τ(\overline{ψ}) ∑_{\begin{matrix}d|m \\ d|k \end{matrix}} ψ (\frac{m}{d}) δ(q, k, d, ψ)  (5.11)

where for a character ψ mod q we define

δ(q, k, d, ψ) = ∑_{\begin{matrix}e|d \\ e| \frac{k}{q} \end{matrix}} \frac{μ(d/e)}{ϕ(k/e)}\overline{ψ} (\frac{−k}{eq} ) ψ (\frac{d}{e}) μ (\frac{k}{eq} ) .  (5.12)

Note that if k is squarefree, then

|δ(q, k, d, ψ)| ≤ ∑_{\begin{matrix}e|d \\ e| \frac{k}{q} \end{matrix}} \frac{ϕ(e)}{ϕ(k)} = \frac{(d, k/q)}{ϕ(k)}.  (5.13)

Now by (5.4) and (5.11), if we replace k by kq and m by md and rearrange, then

M_{v} = ∑_{q≤y} ∑_{ψ}^{∗}τ (\overline{ψ}) ∑_{k≤y/q} \frac{b(kq)}{kq} ∑_{d|kq} δ (q, kq, d, ψ) ∑_{m≤ \frac{kqT}{2πd}} a_{v}(md)ψ(m)

= ∑_{k≤y} \frac{1}{k} N_{v} (\frac{y}{k} , \frac{kT}{2π} , k)  (5.14)

where

N_{ν}(ξ, z, k) = ∑_{q≤ξ} \frac{b(kq)}{q}∑_{ψ}^{∗}τ(\overline{ψ}) ∑_{d|kq} δ(q, kq, d, ψ) ∑_{m≤ \frac{qz}{d}} a_{ν}(md)ψ(m). (5.15)

To estimate this we distinguish the cases q = 1, 1 < q ≤ η = L^{A} for some A > 0, and η < q ≤ ξ ≤ y. The main term arises from q = 1; the case 1 < q ≤ η is treated using Siegel’s Theorem, and the case η > q ≤ ξ depends on an identity for A_{ν} and the large sieve. This is analogous to the proofs of Bombieri’s Theorem given by Gallagher [Ga] and Vaughan [V].

In § 6.1 we consider small values of q > 1; in § 6.2 we consider larger values of q, and in § 7.1 and § 7.2 we evaluate the main terms.

6.1 Small values of q.

By (5.2), and (5.3),

a_{ν}(md) ≪ τ_{4}(m)τ_{4}(d)(1 + (log m)^{4})  (ν = 1, 2).  (6.1.1)

Therefore, by Perron’s formula (Lemma 4) with U = exp((log w)^{\frac{1}{2}} ), T ≪ w ≪ T^{2}, d ≪ T (6.1.2) we have

∑_{m≤w} a_{ν}(md)ψ(m) = \frac{1}{2πi} ∫_{θ−iU}^{θ+iU} A_{ν}(s, ψ, d)w^{s} \frac{ds}{s} + O (τ_{4}(d)\frac{w}{U} L^{C})  (6.1.3)

where θ = 1 + (log w)^{−1}, C > 0 is some constant, and

A_{ν}(s, ψ, d) = ∑_{m=1}^{∞} \frac{a_{ν}(md)ψ(m)}{m^{s}}, σ > 1.  (6.1.4)

We apply Lemma 3 to show that A_{ν}(s, ψ, d) has an analytic continuation to the left of σ = 1. We have by (5.2),

A_{1}(s, ψ, d) = ∑_{d_{1}d_{2}=d} ( ∑_{m=1}^{∞} \frac{Λ(md_{1})ψ(m)}{m^{s}}) ( ∑_{\begin{matrix}m=1 \\ (m,d_{1})=1 \end{matrix}}^{∞} \frac{ψ(m) log md_{2}}{m^{s}} )

= ∑_{d_{1}d_{2}=d} (I(d_{1})\frac{L′}{L} (s, ψ) − \frac{Λ(d_{1})}{1 − ψ(P)P^{−s}} ) d_{2}^{s} \frac{d}{ds} (\frac{L(s, ψ)Φ(s, ψ, d_{1})}{d_{2}^{s}}) (6.1.5)

where

I(d) = \{ \begin{matrix}1 & if d = 1 \\ 0 & if d ≠ 1 \end{matrix} tag6.1.6

Φ(s, ψ, d) = ∏_{p|d} (1 − ψ(p)p^{−s}) = ∑_{n|d} \frac{μ(n)ψ(n)}{n^{s}},  (6.1.7)

and if Λ(d_{1}) ≠ 0 then P = P (d_{1}) is such that Λ(d_{1}) = log P. Similarly, by (5.3),

A_{2}(s, ψ, d) = ∑_{d_{1}d_{2}d_{3}d_{4}=d} (I(d_{1})\frac{L′}{L} (s, ψ) − \frac{Λ(d_{1})}{1 − ψ(P)P^{−s}} )

d_{2}^{s}d_{3}^{s} \frac{d}{ds} (\frac{L(s, ψ)Φ(s, ψ, d_{1})}{d_{2}^{s}})

× \frac{d}{ds} (\frac{L(s, ψ)Φ(s, ψ, d_{1}d_{2})}{d_{3}^{s}}) B(s, ψ, d_{1}d_{2}d_{3}, d_{4}) (6.1.8)

where

B(s, ψ, d, e) = ∑_{\begin{matrix}(m,d)=1 \\ m≤y/e \end{matrix}} \frac{b(me)ψ(m)}{m^{s}}.  (6.1.9)

Thus we see that A_{ν}(s, ψ, d) has a meromorphic continuation to the whole plane. If ψ is the principal character mod 1, then A_{ν}(s, ψ, d) has a pole at s = 1. For any other character, A_{ν}(s, ψ, d) has at most one (simple) pole in the region

σ ≥ σ_{o}(t) := 1 − \frac{c}{log q(1 + |t|)}  (6.1.10)

where c is an absolute constant. By Siegel’s theorem, this pole, if it exists, is at a real number β which satisfies

1 − β ≫_{ε} q^{−ε}  (6.1.11)

for any ε > 0.

We will move the path of integration in (6.1.3) to the segment σ = σ_{o}(U), |t| ≤ U and estimate the integral on the new path. In doing so, we cross the pole at s = 1 if q = 1 and the pole at s = β, if it exists. Let

R_{ν}(w, d) = Res_{s=1} A_{ν}(s, 1, d)\frac{w^{s}}{s} .  (6.1.12)

We note, for future reference, that the contribution of this residue to M_{ν} is, by (5.14),

R_{ν} := ∑_{k≤y} \frac{b(k)}{k} ∑_{d|k} δ(1, k, d, 1)R_{ν} (\frac{kT}{2πd}, d)  (6.1.13)

while by (5.4) and (5.11), it is easy to check that

R_{v} = ∑_{k≤y} \frac{b(k)}{k}R_{v}(k)  (6.1.14)

where

R_{v}(k) := Res_{s=1} A_{ν}(s, −1/k) (\frac{T k}{2π} )^{s} s^{−1}  (6.1.15)

and

A_{ν}(s, h/k) = ∑_{m=1}^{∞} \frac{a_{ν}(m)}{m^{s}}e(mh/k).  (6.1.16)

The latter two formulae will be used in §§ 7.1 and 7.2.

Let Γ be the path consisting of three line segments {σ−iU : θ ≥ σ ≥ σ_{o}}, {σ_{o} + it : − U ≤ t ≤ U}, and {σ + iU : σ_{o} ≤ σ ≤ θ}. Then by (6.1.3), (6.1.12), and Cauchy’s theorem,

∑_{m≤w} a_{ν}(md)ψ(m) − I(q)R_{ν}(w, d)  (6.1.17)

≪ ∫_{Γ} |A_{ν}(s, ψ, d)\frac{w^{s}}{s} ds| + |_{s=β} Res_{s=β} A_{ν}(s, ψ, d)\frac{w^{s}}{s} | + τ_{4}(d)\frac{w}{U} L^{C}.

By (6.1.5), (6.1.8), (6.1.9), and standard estimates,

A_{ν}(s, ψ, d) ≪ τ_{4}(d)L^{C}y^{1−σ_{o}}  (6.1.18)

for σ ≥ σ_{o}(U), |t| ≤ U, |s − 1| ≫ L^{−1}, and |s − β| ≫ L^{−1} where C > 0 is an absolute constant. Moreover, by (6.1.2) and (6.1.10),

w^{σ_{o}(U)} ≪_{A} w exp (−C(A)L^{\frac{1}{2}} )  (6.1.19)

for q ≤ η = L^{A}. With the choice

ε = \frac{1}{2A},  (6.1.20)

it follows from (6.1.11) that

w^{β} ≪_{A} w exp (−C(A)L^{\frac{1}{2}} )  (6.1.21)

for q ≤ η. Hence, by (6.1.17),

∑_{m≤w} a_{ν}(md)ψ(m) − I(q)R_{ν}(w, d) ≪_{A} τ_{4}(d)w exp (−C(A)L^{\frac{1}{2}} )  (6.1.22)

uniformly for T ≪ w ≪ T^{2}, q ≤ η = L^{A} where A > 0 is any fixed constant. We use this in (5.15). Now

|τ(ψ)| = q^{\frac{1}{2}}  (6.1.23)

and by (5.13), if kq is squarefree, then

|δ(q, kq, d, ψ)| ≤ \frac{(d, k)}{ϕ(k)ϕ(q)}.  (6.1.24)

Further, for kQ squarefree, Q ≪ T and σ = \frac{1}{2} or σ = 1 we have

\frac{1}{ϕ(k)} ∑_{d|kQ} τ_{r}(d)d^{−σ}(d, k) =\frac{1}{ϕ(k)} ∑_{d|k} τ_{r}(d)d^{1−σ} ∑_{e|Q} τ_{r}(e)e^{−σ}  (6.1.25)

≪\{ \begin{matrix}τ_{r}(k)k^{−1}L & if σ = 1 \\ τ_{r}(k)k^{− \frac{1}{2}} exp (CL^{\frac{1}{2}} ) & if σ = \frac{1}{2} \end{matrix}

by Lemma 10 and since k ≪ ϕ(k) log log k. Thus, by (5.15) and (6.1.22)– (6.1.25), for w = qz/d and z ≫ T, and any A > 0,

N_{ν} (η, z, k) − b(k) ∑_{d|k} δ (1, k, d, 1) R_{ν} (\frac{z}{d}, d) ≪_{A} z \frac{τ_{r}(k)}{k}L^{C} ∑_{q≤η} q^{\frac{1}{2}} exp (−C (A) L^{\frac{1}{2}} )

≪_{A} τ_{r}(k)\frac{z}{k} exp (−C (A) L^{\frac{1}{2}} )  (6.1.26)

where C(A) denotes a positive function of A not necessarily the same at each occurrence. We now consider the larger values of q.

6.2 Large Sieve estimates.

For larger q we use Perron’s formula (Lemma 4) with U = T^{20}, T ≪ w ≪ T^{2}, and θ = 1 + (log w)^{−1} to obtain

∑_{m≤w} a_{ν}(md)ψ(m) = \frac{1}{2πi} ∫_{θ−iU}^{θ+iU} A_{ν}(s, ψ, d)w^{s} \frac{ds}{s} + O_{ε}(w^{ε}).  (6.2.1)

Then by (5.15), (6.1.24), and (6.1.25),

N_{ν}(ξ, z, k) − N_{ν}(η, z, k) ≪_{ε} max_{Q≤ξ} ∑_{d|kQ} \frac{μ^{2}(kQ)(d, k)}{ϕ(k)} ∫_{η}^{ξ} x^{−1} dS(x) + T^{ε}ξ^{\frac{1}{2}}

where

S(x) = ∑_{q≤x} \frac{q^{\frac{1}{2}}}{ϕ(q)}∑_{ψ}^{∗} |∫_{θ−iU}^{θ+iU} A_{ν}(s, ψ, d) (\frac{qz}{d} )^{s} \frac{ds}{s}|  (6.2.2)

for z ≪ T y. We use an identity for A_{ν} which corresponds to Vaughan’s identity. We have

A_{ν} = H_{ν} + I_{ν}  (ν = 1, 2)  (6.2.4)

where

H_{ν} = H_{ν}(s, ψ, d) = (A_{ν} − F_{ν})(1 − LG)  (6.2.5)

and

I_{ν} = I_{ν}(s, ψ, d) = F_{ν} − F_{ν}LG + A_{ν}LG;  (6.2.6)

here, F_{ν} is a partial sum of A_{ν}, L = L(s, ψ),

F_{ν} = F_{ν}(s, ψ, d) = ∑_{m≤u} \frac{a_{ν}(md)ψ(m)}{m^{s}}  (6.2.7)

and G is a partial sum of L(s, ψ)^{−1},

G = G(s, ψ) = ∑_{m≤v} \frac{μ(m)ψ(m)}{m^{s}}.  (6.2.8)

For the parameters u and v we will take

u = x^{2}, v = T^{\frac{1}{2}} .  (6.2.9)

Now, we have

∫_{θ−iU}^{θ+iU} A_{ν}(s, ψ, d)\frac{w^{s}}{s} ds = ∫_{θ−iU}^{θ+iU}(H_{ν} + I_{ν})(s, ψ, d)\frac{w^{s}}{s} ds.  (6.2.10)

By (6.1.5) and (6.1.8), L(s, ψ)A_{ν}(s, ψ, d) is regular for σ > 0; hence the same is true of I_{ν}. Therefore, by Cauchy’s theorem, if q > 1, then

∫_{θ−iU}^{θ+iU} I_{ν}(s, ψ, d)\frac{w^{s}}{s} ds = ∫_{Γ} I_{ν}(s, ψ, d)\frac{w^{s}}{s} ds  (6.2.11)

SIMPLE ZEROS OF THE RIEMANN ZETA-FUNCTION 21

where Γ is the path consisting of three line segments \{σ − iU : θ ≥ σ ≥ \frac{1}{2}\}, \{\frac{1}{2} +it : −U ≤ t ≤ U\}, and \{σ +iU : \frac{1}{2} ≤ σ ≤ θ\}. We can easily bound the integral along the horizontal segments of Γ. By (6.1.5) and (6.1.8),

A_{ν}(s, ψ, d)L(s, ψ) ≪ ∑_{d_{1}d_{2}d_{3}d_{4}=d}(1 + |L(s, ψ)| + |L′(s, ψ)|)^{3}
× |B(s, ψ)| ∏_{p|d}(1 + p^{−σ})^{2} log^{3} d (6.2.12)

where

B(s, ψ) = ∑_{m≤y} \frac{β(m)ψ(m)}{m^{s}}  (6.2.13)

for some coefficients β which depend on d_{1}, d_{2}, d_{3}, and d_{4}, but satisfy

β(n) ≪ 1  (6.2.14)

uniformly in n and the d_{i}. Thus, for some fixed r,

A_{ν}L ≪ τ_{r}(d)L^{3}(1 + |L| + |L′|)^{3}|B|  (6.2.15)

for d ≪ T and σ ≥ \frac{1}{2}. If 1 < q ≪ T and σ ≥ \frac{1}{2}, then, it is well-known that

L(s, ψ) ≪ (q|s|)^{\frac{1}{4}} ; L′(s, ψ) ≪ (q|s|)^{\frac{1}{4}} L  (6.2.16)

and it is trivial that

F_{ν}(s, ψ, d) ≪_{ε} y^{1+ε}; G(s, ψ), B(s, ψ) ≪ T^{\frac{1}{4}} .  (6.2.17)

Since U = T^{20}, it follows from these estimates and (6.2.11) that

∫_{θ−iU}^{θ+iU} I_{ν}(s, ψ, d)\frac{w^{s}}{s} ds = ∫_{1/2−iU}^{1/2+iU} I_{ν}(s, ψ, d)\frac{w^{s}}{s} ds + O(T^{−1}).  (6.2.18)

Now by (5.15), (6.1.24), (6.1.26), (6.2.2), (6.2.10), and (6.2.18),

N_{ν}(ξ, z, k) − b(k) ∑_{d|k} δ(1, k, d, 1)R_{ν} (\frac{qz}{d} , d)
≪_{A,ε}τ_{r}(k)\frac{z}{k} exp(−C(A)L^{\frac{1}{2}} ) + ξ^{\frac{1}{2}} T^{ε}  (6.2.19)

+ z {max \atop Q ≤ ξ} ∑_{d|kQ} \frac{gμ^{2}(dQ)}{ϕ(k)d} ∫_{η}^{ξ} x^{−1} dH_{v}(x)

+ z^{\frac{1}{2}} {max \atop Q ≤ ξ} ∑_{d|(k)Q} \frac{gμ^{2}(dQ)}{ϕ(k)d^{\frac{1}{2}}} ∫_{η}^{ξ} x^{−1} dI_{v}(x)

where g = (d, k),

H_{v}(x) = ∑_{q≤x} \frac{q^{\frac{3}{2}}}{ϕ(q)}∑_{ψ}^{∗} ∫_{−U}^{U} |H_{v}(θ + it)|\frac{dt}{θ + |t|}  (6.2.20)

and

I_{v}(x) = ∑_{q≤x} \frac{q}{ϕ(q)}∑_{ψ}^{∗} ∫_{−U}^{U} |I_{v}(\frac{1}{2} + it)|\frac{dt}{1/2+it}.  (6.2.21)

To estimate H_{v} and I_{v} we can use Hölder’s inequality and the large sieve inequality in the form

∑_{q≤x} \frac{q}{ϕ(q)}∑_{ψ}^{∗} ∫_{−U}^{U} | ∑_{n} a_{n}ψ(n)n^{−it}|^{2} \frac{dt}{a + |t|}
≪ ∑_{n} (n + x^{2} log U) |a_{n}|^{2}  (6.2.22)

uniformly for a ≥ \frac{1}{2} (See Vaughan [V]).

By (6.1.1), (6.2.7), and (6.2.22),

∑_{q≤x} \frac{q}{ϕ(q)} ∑_{ψ} ∫_{−U}^{U} |A_{v}(θ + it) − F_{v}(θ + it)|^{2} \frac{dt}{θ + |t|}

≪ ∑_{m≥u}(m + x^{2}L)|a_{v}(md)|^{2}m^{−2θ}

≪ (1 + x^{2}u^{−1})L^{C}τ_{4}(d)^{2}  (6.2.23)

for some C. Similarly, by (6.2.8),

∑_{q≤x} \frac{q}{ϕ(q)}∑_{ψ}^{∗} ∫_{−U}^{U} |1 − LG(θ + it, ψ)|^{2} \frac{dt}{θ + |t|}

≪ ∑_{m>1}(m + x^{2}L) |∑_{\begin{matrix}e|m \\ e≤v \end{matrix}} μ(e)ψ(e)ψ (\frac{m}{e})|^{2} m^{−2θ}

≪ (1 + x^{2}v^{−1})L^{C}  (6.2.24)

for some C > 0 since the sum over e is 0 if m ≤ v and is ≪ τ(m) in any case. Hence, by Cauchy’s inequality, (6.2.5), (6.2.9), (6.2.20), (6.2.23) and (6.2.24),

H_{v}(x) ≪ τ_{r}(d)L^{C}(1 + x^{2}u^{−1})^{\frac{1}{2}} x^{\frac{1}{2}}
≪ τ_{r}(d)x^{\frac{1}{2}} (1 + xT^{− \frac{1}{4}} )L^{C}  (6.2.25)

for some C > 0.

Now for an arbitrary f = f(s, ψ) let

A(f) = ∑_{q≤x} \frac{q}{ϕ(q)}∑_{ψ}^{∗} ∫_{−U}^{U} |f(1/2 + it, ψ)| \frac{dt}{\frac{1}{2} + |t|},  (6.2.26)

so that I_{v}(x) = A(I_{v}). By (6.2.6), (6.2.15), and Hölder’s inequality,

I_{v}(x) ≪A(F_{v}^{2})^{\frac{1}{2}} A(1)^{\frac{1}{2}} + A(F_{v}^{2})^{\frac{1}{2}} A(L^{4})^{\frac{1}{4}} A(G^{4})^{\frac{1}{4}}
+ τ_{4}(d)L^{5}A(1 + |L|^{6} + |L′|^{6})^{\frac{1}{2}} A(B^{4})^{\frac{1}{4}} A(G^{4})^{\frac{1}{4}} .  (6.2.27)

By (6.2.7), (6.2.8), (6.2.13), (6.2.22), and (6.1.1),

A(F_{v}^{2}) ≪ (x^{2} + u)L^{C}ε_{r}(d)  (6.2.28)
A(G^{4}) ≪ (x^{2} + v^{2})L^{C},  (6.2.29)

and

A(B^{4}) ≪ (x^{2} + y^{2})L^{C}  (6.2.30)

for some constant C > 0. Also, A(1) ≪ x^{2}L, and by the method of Montgomery [M1] (see also Ramachandra [R]),

A(|L|^{k}), A(|L′|^{k}) ≪_{ε} x^{2}T^{ε} (k = 2 or 4)  (6.2.31)

We also need (6.2.31) with k = 6 which is our sixth moment assumption referred to in the introduction. This follows from GLH and this is the only place we use GLH.

Thus, by (6.2.9), (6.2.27), and the above,

I_{v}(x) ≪_{ε} τ_{4}(d)T^{ε} (x^{1/2}(x^{2} + u)^{1/2}(x^{2} + v^{2})^{1/4} + x(x^{2} + y^{2})^{1/4}(x^{2} + v^{2})^{1/4})
≪_{ε} τ_{4}(d)T^{ε} (x^{2} + x^{3/2}T^{1/4} + x^{3/2}y^{1/2} + xT^{1/4}y^{1/2}) .  (6.2.32)

Next, by (6.2.25),

∫_{η}^{ξ} x^{−1} dH_{v}(x) ≪ (η^{−1/2} + ξ^{1/2}T^{−1/4})τ_{4}(d)L^{C}.  (6.2.33)

By (6.2.32),

∫_{η}^{ξ} x^{−1} dI_{v}(x) ≪_{ε} (ξ + ξ^{1/2}T^{1/4} + ξ^{1/2}y^{1/2} + T^{1/4}y^{1/2})τ_{4}(d)T^{ε}.  (6.2.34)

Hence, by (6.1.25), (6.2.19), and (6.2.32) – (6.2.34),

N_{ν}(ξ, z, k) − b(k) ∑_{d|k} δ(1, k, d, 1)R_{ν} (\frac{dz}{d} , d)

<<_{A,ε} τ_{r}(k) (\frac{z}{k} ) exp (−C(A)L^{1/2}) + τ_{r}(k) (\frac{z}{k} ) (η^{−1/2} + ξ^{1/2}T^{−1/4}) L^{C}
+ τ_{r}(k) (\frac{z}{k} )^{1/2} (ξ + ξ^{1/2}T^{1/4} + ξ^{1/2}y^{1/2} + T^{1/4}y^{1/2})T^{ε}.  (6.2.35)

We use this estimate in (5.14) and by (6.1.13) after substituting ξ = y/k, z = T k/(2π) and summing over k we have,

M_{v} − R_{v} ≪_{A,ε} T exp(−C(A)L^{1/2}) + T η^{−1/2}L^{C} + (T^{3/4}y^{1/2} + T^{1/2}y)T^{ε} (6.2.36)

for some fixed C > 0 and η = L^{A} for A > 0 to be chosen. We take A = 2(A′ + C) and have for any A′ > 0,

M_{v} = R_{v} + O_{A′,ε} (T L^{−A′} + (T^{3/4}y^{1/2} + T^{1/2}y)) T^{ε}  (6.2.37)

This gives M_{v} = R_{v} + o(T) provided y ≪ T^{θ} with θ < 1/2.

The rest of the paper is devoted to estimating the main term.

7. The main term for I.

Recall, by (6.1.15), that

R_{v}(k) = Res_{s=1} A_{v} (s, \frac{−1}{k}) (\frac{kT}{2π} )^{s} \frac{1}{s}.  (7.1)

To evaluate this, we first determine the principal part of A_{v}. For a meromorphic function f(s) let Prin_{s=a} f(s) denote the principal part of f at s = a. Define

p_{1} (\frac{h}{k}) = Prin_{s=1} A_{1} (s, \frac{h}{k})

and

p_{2} (\frac{h}{k}) = Prin_{s=1} A_{3} (s, \frac{h}{k}) ,  (7.2)

where

A_{3} (s, \frac{h}{k}) = ∑_{m=1}^{∞} \frac{a_{3}(m)}{m^{s}}e (\frac{mh}{k})  (7.3)

with

∑_{m=1}^{∞} \frac{a_{3}(m)}{m^{s}} = ζ′(s)^{2} \frac{ζ′}{ζ} (s).  (7.4)

Observe that by (5.3) and (6.1.16),

A_{2} (s, −\frac{1}{k} ) = ∑_{h≤y} \frac{b(h)}{h^{s}}A_{3} (s, −\frac{h}{k}) .  (7.5)

Thus,

R_{ν}(k) = Res_{s=1} ∑_{h≤y} \frac{b(h)}{h^{s}}p_{2}(−h/k) (\frac{kT}{2π} )^{s} \frac{1}{s}.  (7.6)

We shall determine p_{1} and p_{2} by a method due to Estermann [E]. Put

ζ(s, a, k) = ∑_{\begin{matrix}n=1 \\ n≡a(mod k) \end{matrix}}^{∞} n^{−s}

and

ζ(s, h/k) = ∑_{n=1}^{∞} \frac{1}{n^{s}} e (\frac{nh}{k}) .  (7.7)

(The latter notation should not be confused with the Hurwitz zeta-function which will not be used.) Now

Prin_{s=1} ζ(s, a, k) = Prin_{s=1} k^{−s}ζ(s).  (7.8)

Also, ζ (s, \frac{h}{k}) is regular at s = 1 unless k|h, in which case it is equal to ζ(s). Further, define

Λ(s, a, k) = − ∑_{\begin{matrix}n=1 \\ n≡a(mod k) \end{matrix}}^{∞} Λ(n)n^{−s}.  (7.9)

Then if (a,k) = 1 we have

Λ(s, a, k) = − ∑_{n=1}^{∞} \frac{Λ(n)}{n^{s}} \frac{1}{ϕ(k)} ∑_{χ mod k} \overline{χ}(a)χ(n)

= \frac{1}{ϕ(k)} ∑_{χ mod k} \overline{χ}(a)\frac{L′}{L} (s, χ).  (7.10)

Hence

Prin_{s=1} Λ(s, a, k) = Prin_{s=1} \frac{1}{ϕ(k)} \frac{ζ′}{ζ} (s) = \frac{1}{ϕ(k)(1 − s)}  (7.11)

if (a, k) = 1. If (a, k) > 1, then it is easy to see that Λ(s, a, k) is regular for Re s > 0.

7.1 Evaluation of R_{1}(k)

In this subsection we evaluate R_{1}(k) (see (6.1.15)). Now by (5.2) and (6.1.16),

A_{1}(s, −1/k) = ∑_{m=1}^{∞} ∑_{n=1}^{∞} \frac{Λ(m) log n}{m^{s}n^{s}}e (−\frac{mn}{k})

= ∑_{a=1}^{k} Λ(s, a, k)ζ′ (s, −\frac{a}{k} ) .  (7.1.1)

Therefore,

p_{1} (−\frac{1}{k} ) = \{ \begin{matrix}Prin_{s=1} \frac{ζ′}{ζ} (s)ζ′(s) & if k = 1 \\ Prin_{s=1}\{ζ′(s)Λ(s, k, k) + \frac{1}{ϕ(k)} \frac{ζ′}{ζ} (s) ∑_{1}^{k} ′ζ′(s, a/k)\} & if k > 1, \end{matrix}

(7.1.2)

where ∑′ denotes that the sum is restricted to a with (a, k) = 1.

Now, for squarefree k,

Λ(s, k, k) = −Λ(k)(k^{s} − 1)^{−1}  (7.1.3)

and

∑_{a=1}^{k}′ζ′(s, a/k) = \frac{d}{ds} (∑_{n=1}^{∞} \frac{c_{k}(n)}{n^{s}}) = \frac{d}{ds} (ζ(s) ∑_{d|k} μ (\frac{k}{d} ) d^{1−s}) ,  (7.1.4)

where

c_{k}(n) = ∑_{a=1}^{k}′e (\frac{an}{k}) = ∑_{d|(n,k)} dμ (\frac{k}{d} )  (7.1.5)

is Ramanujan’s function (see Titchmarsh [T, p. 10]). The expression in (7.1.4) is regular at s = 1 when k > 1. In fact, it is

\begin{matrix}= \frac{d}{ds} ((\frac{1}{s − 1} + γ + c_{1}(s − 1) + ⋯) \\ (∑_{d|k} μ (\frac{k}{d} ) (1 − (s − 1) log d + \frac{(s − 1)^{2}}{2} log^{2} d + ⋯)))| & s=1 \end{matrix}

= ∑_{d|k} μ (\frac{k}{d} ) \{\frac{1}{2} log^{2} d − γ log d\} = \frac{1}{2}Λ_{2}(k) − γΛ(k),  (7.1.6)

where

Λ_{r} = μ ∗ L^{4}1,

i.e.

Λ_{r}(k) = ∑_{d|k} μ (\frac{k}{d} ) log^{r} d.  (7.1.7)

Thus, it is easy to see that if k is squarefree, then for some constants d_{2} and d_{3},

p_{1} (−\frac{1}{k} ) = \{ \begin{matrix}\frac{1}{(s−1)^{3}} + \frac{d_{2}}{(s−1)^{2}} + \frac{d_{3}}{s−1} & if k = 1 \\ \frac{Λ(k)}{(k−1)(s−1)^{2}} − \frac{1}{s−1} (\frac{kΛ(k) log k}{(k−1)^{2}} + \frac{Λ_{2}(k)/2−γΛ(k)}{ϕ(k)}) & if k > 1. \end{matrix} (7.1.8)

If follows from (7.1), (7.2), and (7.1.8) that for squarefree k,

R_{1}(k) = \{ \begin{matrix}\frac{T}{2π} (\frac{L^{2}}{2} + (d_{2} − 1)L + (d_{3} − d_{2} + 1)) & if k = 1 \\ \frac{kT}{2π} (\frac{Λ(k)}{(k−1)}(L + log k − 1) − \frac{kΛ(k) log k}{(k−1)^{2}} − \frac{(Λ_{2}(k)/2−γΛ(k))}{ϕ(k)}) & if k > 1. \end{matrix}

(7.1.9)

Now, by (2.4), (7.1.7), (5.4), (6.1.14), (6.2.37), and (7.1.9), we have

\begin{aligned}I_{1} & = \frac{T}{2π} (\frac{L^{2}}{2} + O(L) + ∑_{2≤k≤y} b(k)\frac{Λ(k)}{(k − 1)}\{L − 1 − \frac{log k}{k − 1} \\ & − \frac{Λ_{2}(k)}{2ϕ(k)} + \frac{γΛ(k)}{ϕ(k)}\}) + O_{ε}(T^{1/2+ε}y) \end{aligned}

= \frac{T}{2π} (\frac{L^{2}}{2} + ∑_{k≤y} b(k) \{\frac{Λ(k)}{k}L − \frac{Λ_{2}(k)}{2ϕ(k)}\} + O(L))  (7.1.10)
+O_{ε}(T^{1/2+ε}y)

since, for example,

∑_{k≤y} \frac{Λ(k)b(k)}{k} ≪ ∑_{p≤y} \frac{log p}{p} ≪ L.

Next, by the Prime Number Theorem, Stieltjes’ integration and (2.2) – (2.4),

∑_{k≤y} \frac{b(k)Λ(k)}{k} = − ∑_{p≤y} P (\frac{log y/p}{log y}) \frac{log p}{p}
= − (∫_{0}^{1} P (x) dx) log y + O(1).  (7.1.11)

It is easily shown that

Λ_{2} = Λ ∗ Λ + LΛ,

so that

μ(k)Λ_{2}(k) = \{ \begin{matrix}− log^{2} p & if k = p, a prime, \\ 2 log p log q & if k = pq, p ≠ q primes \\ 0 & otherwise. \end{matrix}

This gives

∑_{k≤y} \frac{b(k)Λ_{2}(k)}{ϕ(k)} = − ∑_{p≤y} \frac{log^{2} p}{p − 1}P (\frac{log y/p}{log y})

+ ∑_{pq≤y} \frac{log p log q}{(p − 1)(q − 1)}P (\frac{log y/pq}{log y} ) + O(1)

= O(L),  (7.1.12)

since by Stieltjes’ integration and the Prime Number Theorem, the main terms in (7.1.12) of order L^{2} cancel. Thus

I_{1} = \frac{T}{2π} (\frac{1}{2}L^{2} − ∫_{0}^{1} P (x) dxL log y) + O(T L + T^{1/2+ε}y)

so that by (3.13), we have the required estimate

S_{1} = \frac{T}{2π} (\frac{L^{2}}{2} + L log y ∫_{0}^{1} P (x) dx)) + O_{ε}(T L + T^{1/2+ε}y).  (7.1.13)

28 J. B. CONREY, A. GHOSH, S. M. GONEK

7.2 The Final Calculations

We now consider p_{2}(h/k). First of all write

\frac{h}{k} = \frac{H}{K} , (H, K) = 1.  (7.2.1)

Then one sees by (7.3) that A_{3}(s, h/k) is

= ∑_{a=1}^{K} ∑_{b=1}^{K} Λ(s, a, K)ζ′(s, b, K)ζ′ (s, \frac{abH}{K} )

= ∑_{g|K} ∑_{a=1}^{K/g} ′Λ(s, ag, K) ∑_{b=1}^{K/g} ∑_{c=0}^{g−1} ζ′ (s, b + \frac{cK}{g} , K) ζ′ (s, \frac{abH}{K/g} )

= ∑_{g|K} ∑_{a=1}^{K/g} ′Λ(s, ag, K) ∑_{b=1}^{K/g} ζ′ (s, b, \frac{K}{g} ) ζ′ (s, \frac{abH}{K/g} ) .  (7.2.2)

We employ Estermann’s trick again to see that the above is

= ∑_{g|K} ∑_{a=1}^{K/g} ′Λ(s, ag, K) \{ ∑_{b=1}^{K/g−1} (ζ′ (s, b, \frac{K}{g} ) − \frac{d}{ds} ((\frac{K}{g} )^{−s} ζ(s))) ζ′ (s, \frac{abH}{K/g} )
+ (\frac{d}{ds} (\frac{K}{g} )^{−s} ζ(s)) ∑_{b=1}^{K/g} ζ′ (s, \frac{abH}{K/g} )\},  (7.2.3)

since ζ(s, r, r) = r^{−s} ζ(s) for any positive integer r and where ∑′ denotes a sum over integers a coprime to K/g. Now observe that

∑_{b=1}^{K/g} ζ′ (s, \frac{abH}{K/g} ) = \frac{K}{g} \frac{d}{ds} \{(\frac{K}{g} )^{−s} ζ(s)\} .  (7.2.4)

For r a positive integer define

F (r, s) := ∏_{p|r}(1 − p^{−s}) = 1 ∗ T_{−s}μ(r).  (7.2.5)

Then

∑_{a=1}^{K/g} ′Λ(s, ag, k) = − ∑_{\begin{matrix}a=1 \\ n≡a(modK/g) \end{matrix}}^{K/g}′  ∑_{n=1}^{∞} Λ(ng)(ng)^{−s}
= I(g) (\frac{ζ′}{ζ} (s) + \frac{F′}{F} (\frac{K}{g} , s)) − \frac{Λ(g)}{g^{s} − 1},  (7.2.6)

provided that K is squarefree (so that (g, K/g) = 1). Moreover, the first summand in the curly brackets in (7.2.3) is regular at s = 1. Thus by (7.2.6) and (7.11),

p_{2}(h/k) = Prin_{s=1} K (\frac{ζ′}{ζ} (s) + \frac{F′}{F} (K, s)) \{\frac{d}{ds}K^{−s}ζ(s)\}^{2}

− ∑_{g|K} \frac{Λ(g)}{g^{s} − 1} \frac{K}{g} \{\frac{d}{ds} (\frac{K}{g} )^{−s} ζ(s)\}^{2}

− \frac{1}{ϕ(K)(s − 1)}ζ′ ∑_{a=1}^{K}′ ∑_{b=1}^{K} (s, \frac{abH}{K} )

× (ζ′ (s, b, K) − \frac{d}{ds} \{K^{−s}ζ(s)\}) .  (7.2.7)

We now obtain a different expression for the value at s = 1 of the double sum over a and b. First,

∑_{a=1}^{K}′ ∑_{b=1}^{K} ζ′ (s, \frac{abH}{K} ) = \frac{d}{ds} ∑_{a=1}^{K}′K^{1−s}ζ(s) = ϕ(K)\frac{d}{ds} (K^{1−s}ζ(s)) .  (7.2.8)

Hence, the constant term in

∑_{a=1}^{K}′ ∑_{b=1}^{K} \frac{d}{ds} (K^{−s}ζ(s)) ζ′ (s, \frac{abH}{K} )  (7.2.9)

is the same as the constant term in

Kϕ(K) (\frac{d}{ds}K^{−s}ζ(s))^{2} ,

which is

\frac{ϕ(K)}{K} ∑_{j=0}^{4} α_{j}(log K)^{j}  (7.2.10)

for some constants α_{j}. Next, we determine the constant term in

∑_{a=1}^{K}′ ∑_{b=1}^{K} ζ′(s, b, K)ζ′ (s, \frac{abH}{K} ) .  (7.2.11)

This expression is

= ∑_{a=1}^{K}′ ∑_{m=1}^{∞} ∑_{n=1}^{∞} \frac{log m log n}{(mn)^{s}}c (\frac{mnaH}{K} )

= ∑_{m,n=1}^{∞} \frac{log m log n}{(mn)^{s}}c_{K}(mnH).  (7.2.12)

Note that c_{K}(mnH) = c_{K}(mn) since (H, K) = 1.

Let us now consider the more general expression

C(s, w) = ∑_{m,n=1}^{∞} \frac{log m log n}{m^{s}n^{w}}c_{K}(mn)

= \frac{d^{2}}{ds dw} ∑_{m,n} m^{−s}n^{−w}c_{K}(mn).  (7.2.13)

Now

∑_{m=1}^{∞} m^{−s}c_{K}(mn) = ∑_{d|K} d^{1−s}μ (\frac{K}{d} ) (n, d)^{s}ζ(s)  (7.2.14)

(see Titchmarsh [T, page 10]) and

∑_{n=1}^{∞} \frac{(n, d)^{s}}{n^{w}} = ∑_{g|d} g^{s−w}F (g, s)ζ(w),  (7.2.15)

where F is defined in (7.2.5). Thus,

C(s, w) = \frac{d}{ds} \frac{d}{dw} ζ(s)ζ(w) ∑_{d|K} d^{1−s}μ (\frac{K}{d} ) ∑_{g|d} g^{s−w}F (g, s)

= \frac{d}{ds} \frac{d}{dw} ζ(s)ζ(w) \{μ ∗ T_{1−s} (1 ∗ T_{s−w} (1 ∗ T_{−s}μ))\} (K)

= \frac{d}{ds} \frac{d}{dw} ζ(s)ζ(w) \{μ ∗ T_{1−s}1 ∗ T_{1−w}1 ∗ T_{1−s−w}μ\} (K). (7.2.16)

We perform the differentiations (noting that \frac{d}{ds}T_{s}f(r) = T_{s}Lf(r)) and then let w = s to see that C(s, s) is

= ζ′(s)^{2} {μ ∗ T_{1−s}1 ∗ T_{1−s}1 ∗ T_{1−2s}μ} (K)
− 2ζ(s)ζ′(s){μ ∗ T_{1−s}L1 ∗ T_{1−s}1 ∗ T_{1−2s}μ
+ μ ∗ T_{1−s}1 ∗ T_{1−s}1 ∗ T_{1−2s}Lμ}(K)
+ ζ(s)^{2}{μ ∗ T_{1−s}L1 ∗ T_{1−s}L1 ∗ T_{1−2s}μ
+ 2μ ∗ T_{1−s}L1 ∗ T_{1−s}1 ∗ T_{1−2s}Lμ
+ μ ∗ T_{1−s}1 ∗ T_{1−s}1 ∗ T_{1−2s}L^{2}μ}(K).  (7.2.17)

Therefore, the constant term in C(s, s) at s = 1 is a linear combination of terms of the type

D(α, β, γ; K) := (μ ∗ L^{α}1 ∗ L^{β}1 ∗ T_{−1}L^{γ}μ)(K),  (7.2.18)

with

α + β + γ ≤ 4.  (7.2.19)

We shall use D(K) to denote any possible function of the type described by (7.2.18) and (7.2.19). Thus, by (7.2.7), (7.2.8), (7.2.10), and (7.2.17), we have

p_{2} (\frac{h}{k}) = Prin_{s=1} K^{1−2s}\{(\frac{ζ′}{ζ} (s) + \frac{F′}{F} (K, s)) (ζ′(s) − ζ(s) log K)^{2}

− ∑_{g|K} \frac{Λ(g)}{g^{s} − 1}g^{2s−1} (ζ′(s) − ζ(s) log \frac{K}{g} )^{2}

− \frac{1}{s − 1}(∑_{j=0}^{4} α_{j} log^{j} K + \frac{K}{ϕ(K)} ∑_{i} β_{i}D(K))\}

= Prin_{s=1} K^{1−2s}p_{2}^{∗}(h/k),  (7.2.20)

say, where the β_{i} denote some constants. Now we have by (5.4), (6.1.14), (6.2.37), (7.6), and (7.2.20) that

R_{2} = \frac{T}{2π} ∑_{h,k≤y} \frac{b(h)b(k)(h, k)}{hk} Res_{s=1} (\frac{T}{2πHK} )^{s−1} p_{2}^{∗}(−h/k).  (7.2.21)

The main part of the residue factor above is

= Res_{s=1} (\frac{T}{2πHK} )^{s−1} \{\frac{−1}{s − 1} (\frac{−1}{(s − 1)^{2}} − \frac{log K}{s − 1} )^{2}

− ∑_{g|K} Λ(g)g^{s−1} (\frac{1}{(s − 1)^{2}} + \frac{log K/g}{s − 1} )^{2}\}

= −\frac{L_{0}^{4}}{24} − \frac{L_{0}^{3} log K}{3} − \frac{L_{0}^{2} log^{2} K}{2}

− ∑_{g|K} Λ(g) \{\frac{1}{6}L_{0}^{3} + \frac{1}{2}L_{0}^{2} (log g + 2 log \frac{K}{g} )

+ L_{0}(\frac{log^{2} g}{2} + 2 log g log \frac{K}{g} + log^{2} \frac{K}{g} )
+\frac{log^{3} g}{6} + log^{2} g log \frac{K}{g} + log g log^{2} \frac{K}{g} \}, (7.2.22)

where we have written

L_{0} = log \frac{T}{2πHK} .  (7.2.23)

Now, since

∑_{g|K} Λ(g) = log K,

we can simplify the right hand side of (7.2.22) to

W_{1} : = −\frac{L_{0}^{4}}{24} − \frac{1}{2}L_{0}^{3} log K − \frac{3}{2}L_{0}^{2} log K − L_{0} log^{3} K

− ∑_{g|K} Λ(g) \{−\frac{1}{2}L_{0}^{2} log g − \frac{1}{2}L_{0} log^{2} g + log^{2} K log g

− log K log^{2} g + \frac{1}{6} log^{3} g\}.  (7.2.24)

We now determine the rest of the residue term factor in (7.2.21). It is easy to check that

(\frac{d}{ds})^{n} \frac{F′}{F} (s, K)|_{s=1} = ∑_{p|K} \frac{Q_{n}(p)(log p)^{n+1}}{(p − 1)^{n+1}} ,  (7.2.25)

where Q_{n}(x) is a certain polynomial of degree n. We let F(K) denote any expression such as occurs on the right of (7.2.25) or any linear combination of such, not necessarily the same at each occurrence. Now note that if K is squarefree, we may write (for M = 0, 1, 2, …)

∑_{g|K} Λ(g) \{\frac{g^{2s−1}}{g^{s} − 1} − g^{s−1}\} (log g)^{M} = F(K) + F(K)(s − 1) + ⋯ (7.2.26)

A bit of computation then shows that the residue factor in (7.2.21) is equal to W_{1} plus

W_{2} := ∑_{n=0}^{3} F(K)Q_{n}^{∗}(log K)L_{0}^{3−n} + ∑_{n=0}^{4} α_{n}(log K)^{n} + \frac{K}{ϕ(K)} ∑_{i} β_{i}D(K),
+ ∑_{a+b+c≤2} C_{a,b,c} ∑_{g|K} Λ(g)(log g)^{a}(log K)^{b}L_{0}^{c}  (7.2.27)

where Q_{n}^{∗}(x) denotes a polynomial of degree n. Thus, by (7.2.21), (7.2.22), and (7.2.27),

I_{2} = \frac{T}{2π} ∑_{h,k≤y} \frac{b(h)b(k)}{hk}(h, k)\{W_{1} + W_{2}\} + O_{ε} (T^{1/2+ε}y) .  (7.2.28)

Now we consider the following four types of sums:

S_{1}(M_{1}, M_{2}) = ∑_{h,k≤y} \frac{b(h)b(k)}{hk}(h, k)(log H)^{M_{1}}(log K)^{M_{2}};

S_{2}(M_{1}, M_{2}) = ∑_{h,k≤y} \frac{b(h)b(k)}{hk}(h, k)(log H)^{M_{1}}(log K)^{M_{2}}F_{n}^{∗}(K);

S_{3}(M_{1}, M_{2}, A) = ∑_{h,k≤y} \frac{b(h)b(k)}{hk}(h, k)(log H)^{M_{1}}(log K)^{M_{2}} ∑_{g|K} Λ(g)(log g)^{A};

S_{4}(i) = ∑_{h,k≤y} \frac{b(h)b(k)}{hk}(h, k)\frac{K}{ϕ(K)}D(K).

Here K = k/(h, k) and max{M_{1}, M_{2}, A} ≤ 4. We use the Möbius inversion formula

f((h, k)) = ∑_{\begin{matrix}d|h \\ d|k \end{matrix}} ∑_{ℓ|d} μ(ℓ)f(d/ℓ)  (7.2.29)

to separate the variables h and k. We have

S_{1}(M_{1}, M_{2}) = ∑_{d≤y} \frac{1}{d} ∑_{ℓ|d} \frac{μ(ℓ)}{ℓ}Σ_{1}(d, ℓ, M_{1})Σ_{1}(d, ℓ, M_{2})  (7.2.30)

where

Σ_{1}(d, ℓ, M) = ∑_{k≤y/d} \frac{b(kd)}{k}(log kℓ)^{M};  (7.2.31)

S_{2}(M_{1}, M_{2}) = ∑_{d≤y} \frac{1}{d} ∑_{ℓ|d} \frac{μ(ℓ)}{ℓ}Σ_{1}(d, ℓ, M_{1})Σ_{2}(d, ℓ, M_{2})  (7.2.32)

where

Σ_{2}(d, ℓ, M) = ∑_{k≤y/d} \frac{b(kd)}{k}(log kℓ)^{M}F(kℓ);  (7.2.33)

S_{3}(M_{1}, M_{2}, A) = ∑_{d≤y} \frac{1}{d} ∑_{ℓ|d} \frac{μ(ℓ)}{ℓ}Σ_{1}(d, ℓ, M_{1})Σ_{3}(d, ℓ, M_{2}; A)  (7.2.34)

where

Σ_{3}(d, ℓ, M; A) ∑_{k≤y/d} \frac{b(kd)}{k}(log kℓ)^{M} ∑_{g|kℓ} Λ(g)(log g)^{A};  (7.2.35)

S_{4}(i) = ∑_{d≤y} \frac{1}{d} ∑_{ℓ|d} \frac{μ(ℓ)}{ℓ}Σ_{1}(d, ℓ, 0)Σ_{4}(d, ℓ, i)  (7.2.36)

where

Σ_{4}(d, ℓ, i) = ∑_{k≤y/d} \frac{b(kd)}{k} \frac{kℓ}{ϕ(kℓ)}D(kℓ).  (7.2.37)

We get main terms only from S_{1}(M_{1}, M_{2}) with max{M_{1}, M_{2}} ≤ 1, and from S_{3}(M_{1}, M_{2}, A) with M_{1} ≤ 1.

In the notation of Lemma 5, we have

Σ_{1}(d, ℓ, M) = ∑_{n=0}^{M} \binom{M}{n} (log ℓ)^{M−n}Σ(d, n)  (7.2.38)

so that by Lemma 5,

Σ_{1}(d, ℓ, 0) = \frac{μ(d)d}{ϕ(d)} \frac{P′ (\frac{log y/d}{log y})}{log y} + O_{ε} (L^{−2+ε}(1 + L(d/y)^{b})F_{1}(d)) ,  (7.2.39)

Σ_{1}(d, ℓ, 1) = −\frac{μ(d)d}{ϕ(d)}P (\frac{log y/d}{log y}) + O_{ε} (L^{−1+ε}(log ℓ)(1 + L(d/y)^{b})F_{1}(d)) ,

(7.2.40)

and for M ≥ 2,

Σ_{1}(d, ℓ, M) ≪ Σ(d, M) + (log ℓ)Σ(d, M − 1) + ⋯ + (log ℓ)^{M}Σ(d, 0)(7.2.41)
≪_{ε} L^{M−2+ε}(1 + L(d/y)^{b} + L(log ℓ)^{4})F_{1}(d).

Now d/ϕ(d) ≪ F_{1}(d) and

∑_{ℓ|d} \frac{(log ℓ)^{M}}{ℓ} ≪ F_{1}(d)  (7.2.42)

for bounded M. Therefore, by these estimates and Lemmas 6 and 7,

S_{1}(0, 0) = \frac{1}{log y} ∫_{0}^{1} P′(x)^{2} dx + O_{ε}(L^{−2+ε}),
S_{1}(0, 1) = S_{1}(1, 0) = − ∫_{0}^{1} P (x)P′(x) dx + O_{ε}(L^{−1+ε})
= −1/2 + O_{ε}(L^{−1+ε}),
S_{1}(1, 1) = (log y) ∫_{0}^{1} P (x)^{2} dx + O_{ε}(L^{ε})

and if max{M_{1}, M_{2}} ≥ 2, then

S_{1}(M_{1}, M_{2}) ≪_{ε} L^{M_{1}+M_{2}−2+ε}.  (7.2.43)

Now we are interested in Σ_{2}(d, ℓ, M) where ℓ|d. Since b(kd) = 0 if (k, d) > 1, we may assume that (ℓ, k) = 1 whence, by (7.2.25),

F(kℓ) = F(k) + F(ℓ); F(k) = ∑_{p|k} F(p).

Then, by Lemma 5,

\begin{aligned}Σ_{2}(d, ℓ, M) & = ∑_{n=0}^{M} \binom{M}{n} ((log ℓ)^{M−n}F(ℓ) ∑(d, n) \\ & + ∑_{j=0}^{n} \binom{n}{j} ∑_{p≤ \frac{y}{d}} \frac{F(p)}{p}(log p)^{n−j} ∑(pd, j)) \end{aligned}

≪_{ε} F(ℓ)(1 + log ℓ)^{4}L^{M−1+ε}(1 + L(d/y)^{b})F_{1}(d)  (7.2.44)

since by (7.2.25),

∑_{p≤y} \frac{F(p)(log p)^{n}}{p^{1−b}} ≪ 1

for bounded n. Now

∑_{ℓ|d} \frac{F(ℓ)(log ℓ)^{M}}{ℓ} ≪ F_{1}(d)

so that by (7.2.32) and Lemmas 5, 6, and 7,

S_{2}(M_{1}, M_{2}) ≪_{ε} L^{M_{1}+M_{2}−1+ε}.  (7.2.45)

We now consider Σ_{3}(d, ℓ, M; A). In (7.2.35) the inner sum is = ∑_{g|k} Λ(g)(log g)^{A} + ∑_{g|ℓ} Λ(g)(log g)^{A} = ∑_{g|k} Λ(g)(log g)^{A} + O((log ℓ)^{5})

since we may assume (k, ℓ) = 1 and since ∑_{g|ℓ} Λ(g) = log ℓ. Hence

Σ_{3}(d, ℓ, M; A) = ∑_{g≤y/d} \frac{Λ(g)(log g)^{A}}{g} ∑_{m=0}^{M} \binom{M}{m} (log g)^{M−m}Σ_{1}(dg, ℓ, m)
+ O ((log ℓ)^{5}Σ_{1}(d, ℓ, M)) .  (7.2.46)

By (7.2.39),

Σ_{3}(d, ℓ, 0; A) = ∑_{g≤y/d} \frac{Λ(g)(log g)^{A}}{g} \frac{μ(dg)dg}{ϕ(dg)} P′ (\frac{log y/dg}{log y} ) (log y)^{−1}

+ O_{ε} ( ∑_{g≤y/d} \frac{Λ(g)L^{A}}{g} L^{−2+ε} (1 + L (\frac{dg}{y} )^{b}) F_{1}(d))

+ O((log ℓ)^{5}L^{−1}F_{1}(d));  (7.2.47)

36 J. B. CONREY, A. GHOSH, S. M. GONEK

and by (7.2.39) – (7.2.41), if M ≥ 1,

Σ_{3}(d, ℓ, M; A) = ∑_{g≤y/d} \frac{Λ(g)(log g)^{A+M}}{g} \frac{μ(dg)dg}{ϕ(dg)} P′ (\frac{log \frac{y}{dg}}{log y} ) (log y)^{−1}

− M ∑_{g≤y/d} \frac{Λ(g)(log g)^{A+M−1}}{g} \frac{μ(dg)dg}{ϕ(dg)} P (\frac{log \frac{y}{dg}}{log y} )

+ O_{ε} ( ∑_{g≤y/d} \frac{Λ(g)L^{A+M−2+ε}}{g} (1 + L (\frac{dg}{y} )^{b}) F_{1}(d))

+ O_{ε}((log ℓ)^{5}L^{M−1+ε}F_{1}(d)).  (7.2.48)

Now by Lemmas 8 and 9, and (7.2.47),

Σ_{3}(d, ℓ, 0; A) = − \frac{μ(d)d}{ϕ(d)}A!P^{′(−A−1)} (\frac{log y/d}{log y}) (log y)^{A}
+ O_{ε} ((L^{A−1+ε} + L^{−1} log^{5} ℓ)F_{1}(d)^{2})  (7.2.49)

and for M ≥ 1,

Σ_{3}(d, ℓ, M; A) = − \frac{μ(d)d}{ϕ(d)}(log y)^{A+M}(P^{′(−A−M−1)} (\frac{log y/d}{log y}) (A + M)!

− P^{(−A−M)} (\frac{log y/d}{log y}) (A + M − 1)!M)

+ O_{ε}(F_{1}(d)^{2}(L^{M−1+ε} log^{5} ℓ + L^{A+ε})).  (7.2.50)

The main term in either case M = 0 or M ≥ 1 is

= −\frac{μ(d)d}{ϕ(d)}(log y)^{A+M}P^{(−A−M)} (\frac{log y/d}{log y}) A(A + M − 1)!.

Thus, by (7.2.34), (7.2.39) – (7.2.42), and Lemmas 6 and 7,

S_{3}(0, M, A) = − A(A + M − 1)!(log y)^{M+A} ∫_{0}^{1} P′(x)P^{(−A−M)}(x) dx
+ O_{ε} (L^{M+A−1+ε}) ,  (7.2.51)
S_{3}(1, M, A) = A(A + M − 1)!(log y)^{M+A+1} ∫_{0}^{1} P (x)P^{(−A−M)}(x) dx
+ O_{ε}(L^{M+A+ε}),  (7.2.52)

and

S_{3}(M_{1}, M_{2}, A) ≪_{ε} L^{M_{1}+M_{2}+A−1+ε}  (7.2.53)

if M_{1} ≥ 2.

Now we estimate Σ_{4}(d, ℓ, i) where ℓ|d. By (7.2.37) this involves D(kℓ) where we may assume (k, ℓ) = 1. By (7.2.18),

D(kℓ) = ∑_{u,v} β_{u,v}D(k)D(ℓ)  (7.2.54)

for some constants β_{u,v}. Thus, we first bound

N(d) = ∑_{k≤y/d} \frac{b(kd)}{k} \frac{k}{ϕ(k)}D(k).  (7.2.55)

Now, since k is squarefree,

\frac{k}{ϕ(k)} = ∑_{d|k} 1/ϕ(d) = 1 ∗ 1/ϕ(k)  (7.2.56)

so that

\frac{k}{ϕ(k)}D(k) = (1 ∗ 1/ϕ)(μ ∗ L^{α}1 ∗ L^{β}1 ∗ T_{−1}L^{γ}μ)(k)  (7.2.57)

for some α, β, γ with α + β + γ ≤ 4,

= μ ∗ L^{α}1 ∗ L^{β}1 ∗ f_{γ}(k)  (7.2.58)

where

f_{γ} = T_{−1}L^{γ}μ ∗ μ/ϕ ∗ L^{α} \frac{1}{ϕ} ∗ L^{β} \frac{1}{ϕ} ∗ T_{−1}L^{γ} \frac{μ}{ϕ}.  (7.2.59)

Now suppose, without loss of generality, that α ≥ β. Then by (7.2.31)

N(d) = ∑_{ℓ≤y/d} \frac{μ ∗ L^{β}1 ∗ f_{γ}(ℓ)}{ℓ}Σ_{1}(dℓ, 1, α)  (7.2.60)

and by (7.2.41) this is

≪_{ε} ∑_{ℓ≤y/d} |\frac{μ ∗ L^{β}1 ∗ f_{γ}(ℓ)}{ℓ}| L^{α′+ε}(1 + L(dℓ/y)^{b})F_{1}(d)  (7.2.61)

where α′ = max{0, α − 2}. Now in the notation of (7.1.7), μ ∗ L^{β}1 = Λ_{β} and for θ near 1,

∑_{ℓ≤x} \frac{Λ_{β} ∗ f_{γ}(ℓ)}{ℓ^{θ}} ≪ ∑_{ℓ≤x} \frac{Λ_{β}(ℓ)}{ℓ^{θ}} ∑_{ℓ≤x} \frac{f_{γ}(ℓ)}{ℓ^{θ}} ≪ ∑_{ℓ≤x} \frac{Λ_{β}(ℓ)}{ℓ^{θ}}

since, by (7.2.59), f_{γ} ≪ L^{4}τ_{5}ϕ^{−1} so that f_{γ}(ℓ)ℓ^{−θ} ≪ 1. Thus, by Lemma 9,

N(d) ≪_{ε} L^{α′+β+ε}F_{1}(d) ≪_{ε} L^{2+ε}F_{1}(d).  (7.2.62)

Now, by (7.2.36), (7.2.37), and (7.2.54),

S_{4}(i) ≪ ∑_{d≤y} \frac{1}{d} |∑_{ℓ|d} \frac{μ(ℓ)}{ϕ(ℓ)}D(ℓ)| |Σ_{1}(d, 1, 0)| |N(d)|  (7.2.63)

so that by (7.2.39) and (7.2.62),

S_{4}(i) ≪_{ε} ∑_{d≤y} \frac{F_{1}(d)^{2}}{d} ∑_{ℓ|d} \frac{|D(ℓ)|}{ϕ(ℓ)}L^{1+ε}.  (7.2.64)

We easily see by (7.2.18) that

D(ℓ) ≪ L^{4}τ_{4}(ℓ)

so that

S_{4}(i) ≪_{ε} ∑_{ℓ≤y} \frac{τ_{4}(ℓ) log^{4} ℓ}{ℓϕ(ℓ)} ∑_{d≤y} \frac{F_{1}(d)^{2}}{d} L^{1+ε} ≪_{ε} L^{2+ε}  (7.2.65)

by Lemma 7.

We now complete the evaluation of I_{2} using (7.2.21), (7.2.24), (7.2.27), (7.2.28), (7.2.43), (7.2.45), (7.2.51) – (7.2.53), and (7.2.65). We let S_{M_{1},M_{2}} = S_{1}(M_{1}, M_{2}) and S_{M_{1},M_{2}}(A) = S_{3}(M_{1}, M_{2}; A). Then

∑_{h,k≤y} \frac{b(h)b(k)}{hk}(h, k)L_{0}^{4} = L^{4}S_{0,0} − 8L^{3}S_{0,1} + 12S_{1,1}L^{2} + O_{ε}(L^{2+ε}),

∑_{h,k≤y} \frac{b(h)b(k)}{hk}(h, k)L_{0}^{3} log K = L^{3}S_{0,1} − 3L^{2}S_{1,1} + O_{ε}(L^{2+ε}),

∑_{h,k≤y} \frac{b(h)b(k)}{hk}(h, k) \{L_{0}^{2} log^{2} K + L_{0} log^{3} K\} ≪_{ε} L^{2+ε},

− ∑_{h,k≤y} \frac{b(h)b(k)}{hk}(h, k) ∑_{g|K} Λ(g)

\{ − \frac{1}{2}L_{0}^{2} log g − \frac{1}{2}L_{0} log^{2} g + log^{2} K log g − log K log^{2} g + \frac{1}{6} log^{3} g\}

= \frac{1}{2}\{L^{2}S_{0,0}(1) − 2LS_{0,1}(1) − 2LS_{1,0}(1) − 2S_{1,1}(1) + S_{0,2}(1) + LS_{0,0}(2)

− S_{0,1}(2) − S_{1,0}(2) − 2S_{0,2}(1) + 2S_{0,1}(2) − \frac{1}{3}S_{0,0}(3)\} + O_{ε}(L^{2+ε}),

and

∑_{h,k≤y} \frac{b(h)b(k)}{hk}(h, k) \{∑_{n=0}^{3} F(K)Q_{n}^{∗}(log K)L_{0}^{3−n}

− ∑_{n=0}^{4} α_{n} log^{n} K + \frac{K}{ϕ(K)}Σ_{1}β_{i}D(K)\} ≪_{ε} L^{2+ε}.

Now, by (7.2.51), (7.2.52), and (7.2.53),

−2S_{0,1}(1)L + S_{0,0}(2)L ≪_{ε} L^{2+ε},

2S_{1,1}(1) − S_{1,0}(2) ≪_{ε} L^{2+ε},

and

−S_{0,2}(1) + S_{0,1}(2) − \frac{1}{3}S_{0,0}(3) ≪_{ε} L^{2+ε}.

Therefore,

I_{2} = \frac{T}{2π} \{\frac{L^{4}}{24} S_{0,0} − \frac{L^{3}}{6} S_{0,1} + L^{2}S_{1,1} + \frac{L^{2}}{2} S_{0,0}(1) − LS_{1,0}(1) + O_{ε}(L^{2+ε})\} .

By (7.2.43),

I_{2} = \frac{T}{2π} L^{3}\{ − \frac{∫_{0}^{1} P′(x)^{2} dx}{24θ} + \frac{1}{12} + ∫_{0}^{1} P (x)^{2} dx

− \frac{θ}{2} ∫_{0}^{1} P′(x)P^{(−1)}(x) dx − θ^{2} ∫_{0}^{1} P (x)P^{(−1)}(x) dx\} + O_{ε} (T L^{2+ε}) .

Next,

∫_{0}^{1} P′(x)P^{(−1)}(x) dx = ∫_{0}^{1} P (x) dx − ∫_{0}^{1} P (x)^{2} dx

and

∫_{0}^{1} P (x)P^{(−1)}(x) dx = \frac{1}{2} (∫_{0}^{1} P (x) dx)^{2} .

Therefore,

I_{2} = \frac{T L^{3}}{2π} \{−\frac{∫_{0}^{1} P′(x)^{2} dx}{24θ} + \frac{1}{12} + \frac{3θ}{2} ∫_{0}^{1} P (x)^{2} dx − \frac{θ}{2} ∫_{0}^{1} P (x) dx

−\frac{θ^{2}}{2} (∫_{0}^{1} P (x) dx)^{2}\} + O_{ε} (T L^{2+ε}) .  (7.2.66)

The estimate (2.7) now follows from (3.21), (3.26), (3.27) and (7.2.66). This proves our Theorem.

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